orbit_periods

std.sets.orbit_periods · Level L3

Fixed and periodic points of a self-map f of a finite set {0, …, n − 1}: the minimal period of each x, the least p ≥ 1 with fᵖ(x) = x (0 when x is not periodic: its orbit enters a cycle without returning to x), checked for every smaller power; and the fixed points, period 1. A Scan of orbit_step over n steps, enough on a finite set: every orbit repeats within n steps. Entering a cycle does not make the starting point periodic.

per(x) = min{p ≥ 1 : fᵖ(x) = x}

Signature

orbit_periods(f: i64[n]) → f64[n], f64[n]

Structure

The function as NOVA stores it: one box per input, operation and output, and arrows that carry values. A double border marks another library function this one runs — called once, or by Scan once per element; select it to open that function.

input
operation
constant
call
output

Verification

Signature proven by NOVA’s shape solver, for every size.
Equal to the reference following each orbit in Python until it returns or n steps pass in exact rational arithmetic, on all 40 test cases.
All 300 float64 results are exact: the error is zero.
Interpreter and NumPy backend return bit-identical results.
At float32 too: the 40 test cases rounded to float32, and 40 of them exact, with zero error.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 100%
bit-equal to the NumPy formula in float64: 100%
largest error, in units in the last place: 0

Identity

Calls

orbit_step

Called by

—

sha256:d719d6bee11dfa22e41be9edca38b6443e8c0be43d64e8958ad0642f40ce8e6f

The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.

Control handle

Symbol: Ω:std.sets.orbit_periods · Ω:orbit_periods
Pins: sha256:2f0f9ac84bea3f21c1676fc997aa3e81a69363444696f57835a914a7b1769934this graph and the 1 it reaches through calls
Evidence: sha256:d6699528072b7d2b240427fe1dabe51eb78d31e3c374ab7887461e936e2b270bthe hash of its verification record
Needs: no capability: a pure function

Through NOVA’s control layer, the symbol launches this function only while the program still matches what it pins: a change to this graph, or to any graph it reaches, needs a migration first.